9.1 Introduction

We conclude this monograph by presenting applications of the developed algorithm to common problems in signal processing. We have demonstrated that the MIM algorithm is well suited to problems where the solution is sparse in some domain. To wit, the MIM algorithm yields good solutions to deconvolution problems that are "well structured" and "peaky." Below, we give two applications that leverage this property. First, we show how the MIM algorithm may be used to obtain spectral density estimates with sharper frequency resolution. We then conclude with an extended treatment of an application of the MIM algorithm to blind text restoration, exploiting sparse representations of Roman alphabets.

9.2 Spectral Estimation

Spectral-density estimates are often obtained via the fast Fourier Transform (FFT). Typically, such FFT-based estimates are of low resolution; time-domain windows used to suppress unreliable autocorrelation coefficients also act to blur the spectral estimate. To rectify this loss of resolution, we require robust deconvolution techniques to sharpen the image while still maintaining SNR improvements gained via windowing.

Estimating a unique power spectral density given a truncated version of the autocorrelation function is ill defined, as there may exist an infinite number of spectra consistent with the given data. Well-known methods of spectral estimation, including those exploiting the Wiener-Khintchine equality, typically assume that unknown values of the autocorrelation function are uniformly zero. In an information-theoretic setting such an approach is suspect; the assumption that unknown values are zero infers information about the process that may not be substantiated by the data. In this section, we present a more principled approach to estimation in this setting derived via the mutual information principle (MIP).